These are a simple animated visualizations I created while studying the problem.They contain a sequence of 2 s frames starting at 130 vertices and incrementing 62. 

The red numbered text are the first few starting vertex numbers showing the hexagonal structure of the construction.

On the left is a list of sorted and color coded vertices by the number of edges from a vertex for that frame’s shown vertices (it also lists in the right column the number of vertices with that number of edges & color).

The solution data was obtained from: https://github.com/marijnheule/CNP-SAT

The 510 vertex/2504 edge 5 - color graph solution to the Hadwiger–Nelson problem

The 874 vertex/4461 edge 5 - color graph solution to the Hadwiger–Nelson problem